Questions involving philosophy of mathematics

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14
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4answers
1k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
4
votes
6answers
409 views

Question on induction technique

When one uses induction (say on $n$) to prove something, does it mean the proof holds for all finite values of $n$ or does it always hold when even $n$ takes $\pm\infty$?
7
votes
5answers
156 views

Why do we first introduce the open set definition for continuity instead of the neighborhood definition?

After (nearly) completing my course in topology, something weird just stuck out to me which I hadn't considered before. When first discussing continuity, we often use the following definition: Let ...
0
votes
0answers
15 views

Request for good resources on 'history of infinity' topics [migrated]

Im writing/starting with my bachelor thesis, the subject is about "infinity": what the hell is it, why do we accept it, but most of all my goal is to give an overview of the history of the ...
2
votes
1answer
33 views

Properties of the simplest object in n-dimension

In my boredom, I was thinking about why the simplest 3d object (i.e. the one with the least faces, sides, vertices) was the tetrahedron. After it made sense to me, I realized some cool stuff which was ...
4
votes
2answers
73 views

Why do we focus so much in math on functions (as a subclass of relations)?

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and ...
1
vote
2answers
51 views

Are there undecidable problems for which a solution has been found?

I mean are there examples of problems that have been proven to be undecidable, in the sense that it would not be possible to devise a deterministic computer program that outputs a solution for an ...
0
votes
3answers
35 views

Is ( Set S contains $x$ and only $x$ then does S equals $x$ ) true?

If a set $S$ contains $x$ and only $x$, then does $S$ equal $x$?
3
votes
0answers
64 views

Goldbach's conjecture can't be proved to be undecidable?

Conjectures concerning natural numbers which could be settled by a counterexample can, as far as I understand, not be proved to be undecidable without being proved not having a counterexample at the ...
4
votes
6answers
107 views

the purpose of induction

After getting an answer (in a comment) from peter for this question I have a follow up question. If, in all horses are the same color problem for example, we need to use reason, reason which is ...
1
vote
3answers
158 views

How much of Mathematics is limited by our writing? [closed]

I'm sorry if this question is too vague or otherwise a stupid question. Suppose the mathematicians in some alien civilisation similar to ours sculpted their Mathematics in three dimensions (or ...
1
vote
0answers
45 views

Is there a name for this constant? (0.0100011011…)

It's the simplest number I could think of that contains any finite binary code in its digits: $$\begin{align} c &= 0.0100011011000001010011100101110111...\\ &= ...
2
votes
2answers
344 views

Is Physics really a rigorous subject? [closed]

Though I can't give a precise definition of the term rigor (or better to say mathematical rigor) but intuitively in case of mathematics one may note that when we say that 'the proof is rigorous' we ...
5
votes
0answers
66 views

Apparent Arbitrariness in Mathematics

Something about definitions in mathematics has always interested – confused? - me, I call it “arbitrariness in Mathematics” - it's a bad name, but I don't know a better one. Let me explain: 1st - ...
1
vote
1answer
28 views

maximum number of possible rules of a sequence?

Read this http://www.mathsisfun.com/algebra/sequences-finding-rule.html and also http://en.wikipedia.org/wiki/The_Oxford_Murders_%28film%29 where the scene about the murder note left behind contains a ...
-1
votes
1answer
36 views

A Question Regarding Representing $\mathscr P$($\omega$) as a Digraph and CH

It is well known that one can represent sets as digraphs. What is the proper digraph representation of $\mathscr P$($\omega$)? I ask this because $\mathscr P$($\omega$) is $\Pi_1$ in the Levy ...
1
vote
1answer
46 views

Impredicativity and set theory

I have thought about an example in set theory, but I don't know if its legal to do it, maybe someone can help. Let $\emptyset$ be given and let $A$ be a non empty set. Let us create the subset $X = ...
14
votes
1answer
928 views

What is the “opposite” of the Axiom of Choice?

One might think that, trivially, the "opposite" of AC is $\neg$AC. However, thinking about it differently, I'm not sure this is intuitively the case. AC says that every set has a choice function. ...
3
votes
0answers
32 views

Philosophical implications of P vs NP proof?

Wikipedia article on P vs NP says that "a proof either way would have profound implications for ... Philosophy" without providing further details. So I was wondering what could be the philosophical ...
7
votes
3answers
213 views

Why is the Power Set Operation Inherently Vague?

It is a somewhat common view among mathematicians/philosophers (who have an opinion on the subject) that the power set operation is inherently vague. They go on to say that its inherent vagueness is ...
5
votes
1answer
137 views

What goes wrong in the following argument that our conception of “set” is inconsistent?

This question might sound facetious, but it is a genuine question which I am very much interested in. I apologize in advance if it is too conceptual or philosophical, but I'm optimistic that I might ...
2
votes
1answer
73 views

Can arithmetic truths fix the truth value of the Continuum Hypothesis?

Many logicians and philosophers believe that all sentences expressible in the language of Peano Arithmetic have determinate truth-values, even though no nice formal system can capture all of these ...
3
votes
1answer
81 views

Is the first-order incompleteness of a theory (like arithmetics, set theory or logic itself) avoidable in a second or higher-order axiomatizations?

Can we avoid the first-order incompleteness of a theory (like arithmetics or set theory) in a second-order theory which contains the previous? How does it depend on the chosen semantics or models? If ...
4
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0answers
70 views

Strange Consequences of Large Cardinals in Probability

Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics. On the other hand we know that if we bring even the least ...
0
votes
1answer
127 views

Is Bell's Notion of “Abstract Set” Flawed?

Consider the following definition of "abstract set" given by John L. Bell (who wrote the book "Set Theory: Boolean-valued Models and Independence Proofs") from his preprint "Abstract and Variable ...
2
votes
1answer
75 views

Why the dual of some results are true while others are false?

In mathematics, many results have their "dual" versions. In many cases, if a result is true, then its dual is true as well. However, there are some examples while the dual of a true statement is ...
5
votes
3answers
359 views

Good Sources for Lecture Movies in Set Theory, Logic and Philosophy of Maths

Of course as any other researcher I'm not able to attend any scientific event in my research area. But it is always interesting and useful to watch the lecture movies of these events. I will ...
0
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4answers
116 views

what is $e$ really? what is its meaning? [duplicate]

I don't get it how we came up with $e$ and how can nature use this number so much! that is what I have been told and I only know that $e$ is a specific constant like $\pi$! I understand that $\pi$ ...
1
vote
1answer
68 views

Can a proof be too long? [closed]

Suppose that there is an omnipotent oracle saying that there is a proof for Riemann hypothesis but its proof is so long the universum would collapse before mankind will understand the proof. Would ...
5
votes
0answers
87 views

Are there examples of mathematical problems proven by abduction?

Proof by deduction is a simple principal. For example: All humans are mortal, and Bill is a human; Therefore, Bill is mortal. However, proof by abduction is a bit different. A famous example: ...
2
votes
2answers
50 views

Undecidability and truth

Are there undecidable problems for which a single truth exists? For example, the question about parallels is not decidable from Euclid axioms. But multiple answers are valid and give different kinds ...
0
votes
1answer
171 views

Hand-incalculable Problems

Let's define a "hand-incalculable problem" as a mathematical problem that can not be solved by available human calculation power (using only writing materials and utensils) at a specific date and ...
17
votes
7answers
2k views

Why do we stop at exponentiation stage in arithmetic of natural numbers?

In natural numbers the unary successor operator $S$ is the most natural function which maps each number to the next one. Furthermore we may consider the binary relation $+$ as an iteration of $S$. ...
1
vote
2answers
78 views

How can we explain the discrepancy between $\rightarrow$ (IF-THEN) and $\setminus$ (A-BUT-NOT-B)?

Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain, ordered in the usual way. Then $\mathbb{B}$ is a lattice. It has a join operation $\vee$ that coincides with "OR," a meet operation $\wedge$ that ...
0
votes
1answer
15 views

How could we define the existence of an object/element in the Euclidean space?

Let X be an object/element, What does it mean when I say "X is an object in the Euclidean space"? in other words, What differs an existed object from an unexisted one in the Euclidean space?
3
votes
2answers
72 views

Formal theories dealing with non-commutattive and non-transitive notion of equality

This question is inspired by a philosophical discussion which I don't want to bother you with. As far as I know when we use (or define) the statement "$x$ is equal to $y$" in logic and ordinary ...
0
votes
1answer
43 views

Points and real intervals [closed]

The sorites paradox goes like this: Start with a heap of sand. Remove a grain of sand and you still have a heap; remove another, and another, and another, and you'll still have a heap. Eventually, ...
9
votes
1answer
69 views

Can the set of computable numbers be used as a theoretical basis for calculus?

I recall from my Real Analysis course that the rational numbers $\mathbb{Q}$ are not suitable for doing calculus, and I believe the reason was that $\mathbb{Q}$ does not possess the least-upper-bound ...
0
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0answers
96 views

An adequate difference between $\forall x\in A:P(x)$ and $(\forall x)(x\in A\rightarrow P(x))$?

Ever since I was a young student I have felt doubts about the traditional $(\forall x)$-expression: starting a statement with such an irrational lack of focus doesn't seems reasonable! I mean, all $x$ ...
0
votes
2answers
93 views

Does axiom of foundation/regularity protect against Russell-like paradoxes?

In ZF set theory the axiom of regularity (also called axiom of foundation) says that: In all nonempty sets x there is an element y such that x∩y=∅ As I been told that the intention of the axiom ...
1
vote
1answer
99 views

Does Mathematics exists apart from the mathematician? [closed]

Does Mathematics exists apart from the mathematician? Explain yourself. Mathematics seems to be a projection of the mind. But from where the mind originates? Can the source of the mind be known or you ...
4
votes
2answers
180 views

Are “proofs” that are contingent upon physical reality valid?

Consider the following statement: Let $P$ be any polygon and let $A$ be a point inside of $P$. Then there exists at least one side of $P$ such that the perpendicular from $A$ to said side touches ...
1
vote
2answers
383 views

Is there such a thing as the Fundamental Theorem of ZFC?

I was initially meaning to ask to about a fundamental theorem of mathematics, but the word "mathematics" is very vague. So, my question is the title. I want to know if there is a theorem of ZFC that, ...
2
votes
1answer
98 views

Building math theory on absurd axioms - reducing math to logic

I know similar questions have been asked and i know my terminology might be wrong but I am trying to come to an answer to whether math can be derived from logic. Wikipedia defines logic as use and ...
-2
votes
2answers
210 views

Is Infinity Needed in Maths? Does Infinity Actually Exist? [closed]

I'm asking this question as I have been having an on going online debate with a friend of mine. I claimed that Infinity does in fact exist in Maths and in Reality, as there's a whole plethora of ...
3
votes
5answers
156 views

Is $'' \sum_{n = 1}^{\infty} (-1)^n \; \text{is a real number}''$ an invalid statement or a false proposition?

So we're beginning an introductory logic course and my professor is giving examples for valid statements/ propositions - meaningful statements that are either true or false but not both. So he puts ...
5
votes
2answers
60 views

How is Cartesian coordinate system related to his philosophy

In 1637, Rene Descartes published his famous monograph about philosophy "Discourse on the Method of reasoning well and Seeking Truth in the Sciences", and analytic method of geometry has been come up ...
2
votes
2answers
148 views

Three-valued logic as foundation

Isn't it more natural to use Three-valued logic(false-true-unknown) as the foundation of mathematics? It is a better model for natural languages. And it also can model sentences like the lair paradox ...
2
votes
1answer
27 views

Probability that theoretical results match experimental results

I am not sure if this can be determined, but I was wondering if there was any way to go deeper into probability to find the odds that your experimental results match your theoretical results. For ...
8
votes
4answers
202 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...